We describe the missing class of the hierarchy of mixed unit interval graphs,
generated by the intersection graphs of closed, open and one type of half-open
intervals of the real line. This class lies strictly between unit interval
graphs and mixed unit interval graphs. We give a complete characterization of
this new class, as well as quadratic-time algorithms that recognize graphs from
this class and produce a corresponding interval representation if one exists.
We also mention that the work in arXiv:1405.4247 directly extends to provide a
quadratic-time algorithm to recognize the class of mixed unit interval graphs.Comment: 17 pages, 36 figures (three not numbered). v1 Accepted in the TAMC
2015 conference. The recognition algorithm is faster in v2. One graph was not
listed in Theorem 7 of v1 of this paper v3 provides a proposition to
recognize the mixed unit interval graphs in quadratic time. v4 is a lot
cleare